Good rational approximants to the Thue-Morse constant (part 1)

The Thue-Morse sequence is produced by an automaton that starts from the single symbol 0 and successively replaces 0 with the string 01 and 1 with the string 10. Treating the resulting sequence as a binary expansion gives the Thue-Morse constant. Good rational approximants of this number (or any number) are found by creating its continued fraction expansion. It was recently shown, by Badziahin and Zorin, that the partial quotients in this continued fraction are unbounded, so that there are rational approximations whose distance to the number is less than any constant divided by the square of the denominator of the approximant. That result will be treated in a later talk. In this talk, an earlier construction, by Bugeaud and Queffélec in 2012, of sequences of good approximants having a common pattern will be described.

R. T. Bumby